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Electronic Transactions on Numerical Analysis.
Volume 40, pp. 148-169, 2013.
Copyright  2013, Kent State University.
ISSN 1068-9613.
Kent State University
http://etna.math.kent.edu
TOWARD AN OPTIMIZED GLOBAL-IN-TIME SCHWARZ ALGORITHM
FOR DIFFUSION EQUATIONS WITH DISCONTINUOUS AND
SPATIALLY VARIABLE COEFFICIENTS.
PART 1: THE CONSTANT COEFFICIENTS CASE∗
FLORIAN LEMARIɆ, LAURENT DEBREU†, AND ERIC BLAYO‡
Abstract. In this paper we present a global-in-time non-overlapping Schwarz method applied to the onedimensional unsteady diffusion equation. We address specifically the problem with discontinuous diffusion coefficients, our approach is therefore especially designed for subdomains with heterogeneous properties. We derive
efficient interface conditions by solving analytically the minmax problem associated with the search for optimized
conditions in a Robin-Neumann case and in a two-sided Robin-Robin case. The performance of the proposed schemes
are illustrated by numerical experiments.
Key words. optimized Schwarz methods, waveform relaxation, alternating and parallel Schwarz methods
AMS subject classifications. 65M55, 65F10, 65N22, 35K15
1. Introduction. Numerous geophysical phenomena with a strong societal impact involve the coupled ocean-atmosphere system; e.g., those for climate change, tropical cyclones,
or sea-level rise predictions. To get a good depiction of the complex air-sea dynamics, it is
often necessary to couple atmospheric and oceanic simulation models. However, connecting
the two model solutions at the air-sea interface is a difficult problem, which is presently often
addressed in a simplified way from a mathematical point of view. Indeed, with the ad-hoc
coupling methods currently in use, the fluxes exchanged by the two models are generally not
in exact balance [17]. This may be one factor explaining the generally observed strong sensitivity of coupled solutions to the initial conditions or parameter values [23]. This kind of
coupling raises a number of challenges in terms of numerical simulation since we are considering two highly turbulent fluids with widely different scales in time and space. It is thus
natural to use some specific numerical treatment to match the physics of the two fluids at their
interface. It is known that, even if numerical models are much more complicated, a simple
one-dimensional diffusion equation is relevant to locally represent the turbulent mixing in the
boundary layers encompassing the air-sea interface. The corresponding diffusion coefficients
are spatially variable and their values are given by a so-called eddy-viscosity closure [21].
To perform this coupling in a more consistent way than ad-hoc methods, we propose here to
adapt a global-in-time domain decomposition based on an optimized Schwarz method. This
type of method is thoroughly described in [9] and designed thanks to the pioneering work
in [12, 14]. Schwarz-like domain decomposition methods provide flexible and efficient tools
for coupling models with non-conforming time and space discretizations [3, 10]. Transmission conditions of Robin type have been proposed in [19] to circumvent the inability of the
classical Schwarz method (i.e., with the exchange of Dirichlet data) to solve coupling problems in the case of non-overlapping subdomains. Then, thanks to the free parameters in the
Robin conditions, an optimization of the convergence speed has been proposed in [12, 15]:
this is the basis of the so called optimized Schwarz methods (OSM). This kind of method, originally introduced for stationary problems, has been extended to the unsteady cases by adapt∗ Received September 29, 2010. Accepted March 11, 2013. Published online on July 5, 2013. Recommended by
Martin Gander.
† INRIA Grenoble Rhône-Alpes, Montbonnot, 38334 Saint Ismier Cedex, France and Jean Kuntzmann Laboratory, BP 53, 38041 Grenoble Cedex 9, France, ({florian.lemarie, laurent.debreu}@inria.fr).
‡ University of Grenoble, Jean Kuntzmann Laboratory, BP 53, 38041 Grenoble Cedex 9, France
(eric.blayo@imag.fr).
148
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OPTIMIZED SCHWARZ ALGORITHM FOR DIFFUSION EQUATIONS I
149
ing the waveform relaxation algorithms to provide a global-in-time Schwarz method [14, 16]
(sometimes referred to as Schwarz waveform relaxation). This notion of optimization of the
convergence speed is critical in the context of ocean-atmosphere coupling as the numerical
codes involved are very expensive from a computational point of view. In the present series of
two papers, we derive interface conditions leading to an efficient Schwarz coupling algorithm
between two unsteady diffusion equations defined on non-overlapping subdomains. The convergence properties of this kind of problem have already been extensively studied in the case
of a constant diffusion coefficient having the same value in all subdomains [8]. There are
some asymptotic results in the case of coefficients with different constant values in the different subdomains [10] (in the more general case of advection-diffusion-reaction equations). In
the present papers, we extend these studies to the general case of diffusion coefficients which
vary in each subdomain and whose values are different on both sides of the interface. In
this first part, we consider the case of diffusion coefficients that do not vary spatially in each
medium. We study a zeroth-order two-sided optimized method by considering two different
Robin conditions on both sides of the interface. In the second paper [18], the emphasis is on
the impact of the spatial variability of the coefficients on the convergence speed.
This first paper is organized as follows. In Section 2, we recall the basics of optimized
Schwarz methods in the framework of time evolution problems. Sections 3 and 4 are dedicated to the study of a diffusion problem with discontinuous but piecewise constant coefficients. In Section 3, we analytically determine the solution of an optimization problem to
improve the convergence speed of a simplified algorithm with only one Robin condition combined with a Neumann condition. In Section 4, we address the more general case of two-sided
optimized Robin-Robin transmission conditions determined through a thorough study of the
behavior of the convergence factor. Finally in Section 5, some numerical results are shown to
prove the efficacy of the optimized algorithms derived in the previous sections.
2. Model problem and optimized Schwarz methods. Our guiding example is the onedimensional diffusion equation of a scalar u
(2.1)
Lu = ∂t u − ∂x (D(x)∂x u) = f
in Ω × [0, T ],
where Ω is a bounded domain defined as Ω =] − L1 , L2 [, (L1 , L2 ∈ R+ ) and D(x) > 0,
for x ∈ Ω. In practical applications, L1 denotes the bottom of the ocean (of the order of 5 km
in the open ocean), while L2 is typically the top of the troposphere (of the order of 15 km).
This problem is supplemented by an initial condition
u(x, 0) = u0 (x),
x ∈ Ω,
and boundary conditions
B1 u(−L1 , t) = g1 ,
B2 u(L2 , t) = g2 ,
t ∈ [0, T ],
where B1 and B2 are two partial differential operators. In this paper, we always assume
that u0 ∈ H 1 (Ω), f ∈ L2 (0, T ; L2 (Ω)) and that D(x) is bounded in the L∞ -norm. Note
that in actual applications such assumptions are generally fulfilled. Existence and uniqueness
results for this problem can be proved following [10] and are not discussed here.
2.1. Formulation of the global-in-time Schwarz method. In the present study, we
consider a case where the diffusion coefficient D(x) has one discontinuity in Ω. This discontinuity represents the transition between two media with heterogeneous physical properties. In this case we define two subdomains, each subdomain having its own diffusion
profile Dj (x), (j = 1, 2). This amounts to splitting Ω into two non-overlapping domains Ω1
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F. LEMARIÉ, L. DEBREU, AND E. BLAYO
F IG . 2.1. Decomposition of the spatial domain Ω into two non-overlapping subdomains.
and Ω2 ; see Figure 2.1. These subdomains communicate through their common interface
at Γ = {x = 0}. (Note that there can be various reasons for such a splitting: different physics,
parallelization and/or different numerical treatment requirements.) We propose to use a nonoverlapping global-in-time Schwarz algorithm to solve the corresponding coupling problem.
This method consists in iteratively solving subproblems in Ω1 × [0, T ] and Ω2 × [0, T ] using
the values computed at the previous iteration in the other subdomain as an interface condition
at x = 0. The operator L introduced in (2.1) is split into two operators Lj = ∂t −∂x (Dj (x)∂x )
restricted to Ωj (j = 1, 2). Introducing the operators F1 , F2 , G1 , and G2 to define the interface
conditions, the algorithm reads
(2.2)
L1 uk1
k
u1 (x, 0)
=
=
=
=
f,
uo (x),
g1 ,
F2 u2k−1 (0, t),
in Ω1 × [0, T ],
x ∈ Ω1 ,
t ∈ [0, T ],
in Γ × [0, T ].
L2 uk2
k
u2 (x, 0)
B2 uk2 (L2 , t)
G2 uk2 (0, t)
=
=
=
=
f,
uo (x),
g2 ,
G1 uk1 (0, t),
in Ω2 × [0, T ],
x ∈ Ω2 ,
t ∈ [0, T ],
in Γ × [0, T ],
B1 uk1 (−L1 , t)
F1 uk1 (0, t)
where k = 1, 2, ... is the iteration number, and where the initial guess u02 (0, t) is given. Algorithm (2.2) corresponds to the so-called “multiplicative” form of the Schwarz method. If
we replace the interface condition G2 uk2 = G1 uk1 on Ω2 by G2 uk2 = G1 u1k−1 , we obtain the
“parallel” version of the algorithm. The multiplicative form converges more rapidly than the
parallel one but prevents us from solving subproblems in parallel (this problem can, however,
be circumvented when we consider more than two subdomains). The interested readers may
refer to [7] for further details regarding the different variants of the Schwarz method. Although the present study uses the multiplicative form of the algorithm, the theoretical results
regarding the determination of optimized transmission conditions are also valid for the parallel form. Note that the usual algorithmic approach used in ocean-atmosphere climate models
as described in [4] generally corresponds to one (and only one) iteration of the algorithm
in (2.2) (with Fj = Gj = Dj (0)∂x , j = 1, 2).
The primary role of the operators Fj and Gj (j = 1, 2) in (2.2) is to ensure a given
consistency of the solution on the interface Γ. In our context we require the equality of
the subproblems solutions and of their fluxes. The most natural choice to obtain such a
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connection consists in choosing
F1 = D1 (0)
∂
,
∂x
F2 = D2 (0)
∂
,
∂x
and G1 = G2 = Id.
However, as proposed in [19], the same consistency can be obtained using mixed boundary
conditions of Robin type, leading to
(2.3)
Fj = Dj (0)
∂
+ Λ1 ,
∂x
and Gj = Dj (0)
∂
+ Λ2
∂x
(j = 1, 2).
The advantage of (2.3) is that if the operators Λ1 and Λ2 are correctly chosen, then we can
greatly improve the convergence speed of the corresponding algorithm [12]. Note that Λj
must also be carefully chosen to ensure the well-posedness of the problem. In this paper
we focus on Robin-type transmission conditions since Dirichlet-Neumann-type algorithms
generally converge quite slowly except for large discontinuities between the coefficients D2
and D1 . (It can easily be shown that the convergence rate is given by the square root of the
ratio between D1 and D2 .)
At this point, we have formulated the coupling problem we want to address. The convergence properties of this kind of problem have been extensively studied in the case of
constant and continuous diffusion coefficients [8]. There are also some results in the case
of constant and discontinuous coefficients [10] in the more general case of an advectiondiffusion-reaction problem. This latter study provides results for specific asymptotic cases
that are discussed later in Section 4.4. In this paper, we propose to investigate the problem with diffusion coefficients being constant in each subdomain and discontinuous at the
interface, i.e., Dj (x) = Dj , with Dj > 0 and D1 6= D2 . We prove convergence of the algorithm (2.2) and we determine optimal choices for the operators Λj under some constraints on
the parameters of the problem.
2.2. Convergence of the algorithm. A classical approach to demonstrate the convergence of algorithm (2.2) consists of introducing the error ekj between the exact solution u⋆ and
the iterates ukj , j = 1, 2. By linearity, those errors satisfy homogeneous diffusion equations
with homogeneous initial conditions. We denote the Fourier transform in time by gb = F(g)
for any g ∈ L2 (R). Assuming that T → ∞ and that all the functions are equal to zero for
negative times, it can easily be shown that the errors ebkj in Fourier space satisfy a second-order
ordinary differential equation in x
∂ 2 ebkj
=0
for x ∈ Ωj , ω ∈ R∗ ,
∂x2
q
|ω|
|ω|
with characteristic roots σj± = ± 2D
i
. Note that the particular case ω = 0
1
+
ω
j
would correspond to the existence of a stationary part in the error. However, since the error
is initially zero, such a stationary part is also necessarily zero. To study the convergence of
algorithm (2.2), it is usually assumed that L1 , L2 → ∞, thus leading to
iωb
ekj − Dj
(2.4)
ebk1 (x, ω)
ebk2 (x, ω)
=
=
+
αk (ω)eσ1 x ,
−
β k (ω)eσ2 x ,
for x < 0, ω ∈ R∗ ,
for x > 0, ω ∈ R∗ .
The validity of this assumption is discussed in [17]. The functions α(ω) and β(ω) are determined using the Robin interface conditions at x = 0
(2.5)
( D1 σ1+ + λ1 )αk (ω)
(−D2 σ2− + λ2 )β k (ω)
=
=
( D2 σ2− + λ1 )β k−1 (ω),
(−D1 σ1+ + λ2 )αk (ω),
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where λj is defined as the symbol of the operator Λj (j = 1, 2). A convergence factor ρ of
the Schwarz algorithm (2.2) can be defined as
k
eb1 (0, ω) ebk2 (0, ω) =
.
ρ(ω) = k−1
eb1 (0, ω) eb2k−1 (0, ω) Given (2.4), this amounts to ρ(ω) = αk /αk−1 = β k /β k−1 . Using (2.5) we obtain
(2.6)
(λ1 (ω) + D2 σ2− ) (λ2 (ω) − D1 σ1+ ) .
ρ(ω) = (λ1 (ω) + D1 σ1+ ) (λ2 (ω) − D2 σ2− ) A more general derivation of the convergence factor for the case of an advection-diffusionreaction problem with discontinuous coefficients can be found in [10]. At this point, we are
not able to infer the convergence (or the divergence) of the corresponding algorithm because
the operators Λj have not been explicitly determined. It is often a difficult task to choose
them in an appropriate way. The main difficulty comes from the fact that the convergence
factor is formulated in the Fourier space, meaning that we can only act on symbols λj and
not directly on pseudo-differential operators Λj in the physical space.
2.3. The optimized Schwarz method. It is possible to find values λ1 and λ2 canceling
the convergence factor (2.6) and therefore ensuring a convergence in exactly two iterations.
Their expressions are
r
r
|ω|
|ω|
|ω|D2
|ω|D1
opt
opt
+
−
(1 +
i) and λ2 = D1 σ1 =
(1 +
i).
(2.7) λ1 = −D2 σ2 =
2
ω
2
ω
These symbols correspond to so-called absorbing conditions. Unfortunately, since these optimal symbols are not polynomials in iω, the absorbing conditions are nonlocal in time in the
physical space. The problem is thus to determine local operators providing a good approximation of nonlocal ones by finding a polynomial form in iω to approximate λopt
j . There are
mainly two approaches for such an approximation [12]. The first approach is a low frequency
approximation, namely a Taylor expansion for a small ω. We decided not to adopt this approach because we want to be able to consider a wide range of frequencies. The second and
more sophisticated approach is to solve a minmax problem to determine local operators that
optimize the convergence speed over the full range of admissible frequencies [ωmin , ωmax ].
For a zeroth-order approximation, we look for values λ0j ∈ R such that λ0j ≈ λopt
j . The num0
bers λj can be defined as the solution of the optimization problem
(2.8)
min
0
0
λ1 ,λ2 ∈R
max
ω∈[ωmin ,ωmax ]
ρ(λ01 , λ02 , ω) .
Since we work on a discrete problem, the frequencies allowed by our temporal grid range
π
from ωmin = Tπ to ωmax = ∆t
, where ∆t is the time step of the temporal discretization. The
analytical solution of problem (2.8) is not an easy task: the minimization of a maximum
is known to be one of the most difficult problems in optimization theory [5]. Moreover,
we are faced with an optimization problem for two parameters λ01 and λ02 , which substantially enlarges the difficulty. Some analytical results exist in the case of a two-sided optimization for the 2D stationary diffusion equation [6, 20] and for the 2D Helmholtz equation [11]. In [10], the asymptotic solution of (2.8) for an advection-diffusion-reaction problem for ∆t → 0, ωmin = 0, and a positive advection is found in two particular cases: first
for λ01 = λ02 (one-sided) and second for λ01 6= λ02 (two-sided) but D1 = D2 . In this paper, we
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study the complete minmax problem (2.8) in the general case λ01 6= λ02 and D1 6= D2 . Solving numerically the minmax problem (2.8) is quite expensive from a computational point of
view. Moreover, this optimization must be performed for any change in the values of D1 and
D2 . That is why we look for an analytical solution in the case of a zeroth-order approximation of the absorbing conditions. This is done with two different sets of interface conditions,
the Neumann-Robin case and the Robin-Robin case.
Algorithm (2.2) with two-sided Robin conditions is well-posed for any choice of λ01 and
0
λ2 such that λ01 + λ02 > 0. This result can be shown following the methodology based on a
priori energy estimates as described in [1] and [8].
3. The optimized Schwarz method with Neumann-Robin interface conditions. In
this section, we assume that the solution in Ω2 is subject to a Neumann boundary condition.
The convergence speed of the Neumann-Robin algorithm is expected to be slower than that
one obtained by a Robin-Robin algorithm. However, this easier case is treated explicitly because it introduces several methodological aspects useful for the determination of the general
Robin-Robin optimized interface conditions. Imposing a Neumann boundary condition on
the solution u2 on Γ corresponds to having Λ2 = 0 in (2.3). The convergence factor ρNR (NR
stands for “Neumann-Robin”) obtained from (2.6) reduces to
D1 σ1+ (D2 σ2− + λ1 ) .
(3.1)
ρNR = D2 σ2− (D1 σ1+ + λ1 ) T HEOREM 3.1 (Optimized Robin parameter). The analytical solution λ0,⋆
1 of the minimax problem
0
max
ρNR (λ1 , D1 , D2 , ω)
min
0
λ1 ∈R
ω∈[ωmin ,ωmax ]
is given by
λ0,⋆
1
(
p
p √
√
1
= √
D2 − D1 ( ωmin + ωmax )
2 2
)
r
p
p 2 √
p
√
√
2
D2 − D1 ( ωmin + ωmax ) + 8 D1 D2 ωmin ωmax .
+
Proof. Introducing ζ =
p
|ω|D1 , γ =
p
D2 /D1 , λ01 =
p
ζmin ζmax /2 p, for p ∈ R,
and making σ1+ and σ2− in (3.1) explicit, we obtain
s
1 (p − γζ)2 + γ 2 ζ 2
ρNR (p, ζ) =
,
γ
(p + ζ)2 + ζ 2
p
with ζ = ζ/ ζmax ζmin . Moreover, to ensure the well-posedness ofpthe algorithm, we consider λ01 > 0, i.e., p > 0. Defining an additional parameter µ = ζmax /ζmin , we obtain
that ζ is bounded by ζ min = µ−1 and ζ max = µ. The aim is to optimize the convergence
speed by finding p⋆ , the solution of the minimax problem
max ρNR (p, ζ) .
min
p>0
ζ∈[µ−1 ,µ]
We first study the behavior of the derivative of ρNR with respect to ζ and p with ζ ≥ 0
p
and p ≥ 0. For simplicity we introduce q = p/ γ − 1 + 1 + γ 2 .
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We first derive some restriction of the parameter’s range. We can easily show that
∂ρNR
(3.2)
Sign
= Sign (q − ζ) .
∂p
Looking at the sign of the derivative of ρNR with respect to p, we see that for all values
of ζ, the convergence factor ρNR is a decreasing function of p for q < ζ min = µ−1 , proving
that q ⋆ ≥ ζ min . A similar argument shows that q ⋆ ≤ ζ max . This proves that the optimized
parameter q ⋆ satisfies
1/µ ≤ q ⋆ ≤ µ.
Along with (3.2), this shows that the convergence factor has to be an increasing function of p
at ζ = 1/µ and a decreasing function of p at ζ = µ.
Next we show an equioscillation property of the optimal parameter. The sign of the
derivative of ρNR with respect to ζ is given by
∂ρNR
= Sign (ζ − q) .
Sign
∂ζ
This relation implies that ρNR has a local minimum between 1/µ and µ. The maximum
value of the convergence factor is thus attained either at ζ = 1/µ or at ζ = µ (or both).
If we assume ρNR (p, 1/µ) < ρNR (p, µ), it is always possible to decrease the maximum value
of ρNR (p, ζ) by increasing the value of p so that we have ρNR (p, 1/µ) ≥ ρNR (p, µ). A similar argument shows that ρNR (p, µ) ≥ ρNR (p, 1/µ). The optimal parameter must thus satisfy
the equioscillation property ρNR (p⋆ , 1/µ) = ρNR (p⋆ , µ). After some simple algebra, we find
that p⋆ is a solution of
(γ − 1) (µ + 1/µ) +
2γ
− p⋆ = 0.
p⋆
⋆
⋆
The unique positive solution of the equation v ⋆ = 2γ
p⋆ − p with v = (1 − γ) (µ + 1/µ) is
p
given by p⋆ = 12 −v ⋆ + 8γ + (v ⋆ )2 . After a substitution of γ and µ and a multiplication
p
of p⋆ by ζmin ζmax /2, we retrieve the stated result for λ0,⋆
1 .
We find that the optimized convergence factor satisfies an equioscillation property. This
concept of equioscillation property comes from the Chebyshev’s alternation theorem (or
equioscillation theorem). The similarities between the Chebyshev’s theorem and the optimized Schwarz method are clearly exposed in [2, 6]. Two typical optimized convergence
factors ρ⋆NR = ρNR (λ0,⋆
1 ) are shown in Figure 3.1 (left) for µ = 2 and µ = 6 with γ = 5. Note
that the performance of the optimized algorithm is only a function of the ratio γ between D1
and D2 and not of the actual values of those parameters. The same remark applies to the
temporal frequencies ωmin and ωmax : ρ⋆NR is only a function of their ratio µ.
It is also instructive to look at three particular cases: γ → 0+ , γ = 1, and γ → ∞.
• γ → 0+ (D1 ≫ D2 ):
s
2
1/4
ωmax
µ
0,⋆
,
lim
λ
=
0,
with
µ
=
.
lim+ ρ⋆NR = 1 − 2
1
1 + µ2
ωmin
γ→0+
γ→0
The√minimum value of the convergence factor is attained at µ = 1 and is equal
to 2/2. When µ is increased, the convergence is very slow. Indeed, we tend
towards a Neumann-Neumann algorithm in this case.
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F IG . 3.1. Behavior of ρNR (λ0,⋆
1 ) with respect to ω for γ = 5, µ = 2, and µ = 6 (left). Optimized convergence
factor as a function of γ for µ = 2 and µ = 6 (right).
• γ = 1 (D1 = D2 = D):
s
⋆
ρNR =
√
2 2µ
√ ,
1−
1 + µ(µ + 2)
λ0,⋆
1 =
√
D (ωmax ωmin )
1/4
.
The convergence factor ρ⋆NR approaches 1 when µ is increased. One can also remark
that the optimal parameter λ0,⋆
1 is the same as that one found in [8] in the RobinRobin one-sided case.
• γ → +∞ (D1 ≪ D2 ):
lim ρ⋆NR = 0,
γ→+∞
lim λ0,⋆
1 = +∞.
γ→+∞
When γ tends to +∞, the convergence is very fast (the convergence factor approaches 0) and the optimal boundary condition tends towards a Neumann-Dirichlet
operator.
These results are illustrated in Figure 3.1 (right). The efficiency of the Neumann-Robin algorithm is greatly improved when γ becomes large and µ becomes small. We continue this
section by studying the asymptotic convergence rate for the discretized algorithm when the
time step ∆t tends to 0.
π
T HEOREM 3.2 (Asymptotic performance). For D2 > D1 (i.e., γ > 1), ωmax = ∆t
and
for ∆t tending to zero, the optimal Robin parameter given by Theorem 3.1 is
p
γ − 1√
γ2 + 1 √
−1/2
2D
π∆t
+
ω
λ0,⋆
≈
1
min
1
2
2(γ − 1)
and the asymptotic convergence of the optimized Neumann-Robin algorithm is
r
(γ + 1) ωmin 1/2
1
0,⋆
max π ρNR (λ1 , ω) =
1−
+ O(∆t).
∆t
ωmin ≤ω≤ ∆t
γ
(γ − 1)
π
We conclude that the zeroth-order optimized Neumann-Robin boundary conditions are efficient when the Robin condition is imposed at the boundary of the domain with the smaller
⋆
diffusion
p coefficient (Ω1 here).
In this case, the asymptotic convergence factor ρNR is of the
1/2
form D1 /D2 1 − O(∆t ) for small ∆t. In the next section, we study the zeroth-order
two-sided Robin-Robin boundary conditions.
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4. The optimized Schwarz method for a diffusion problem with discontinuous (but
constant) coefficients: two-sided Robin transmission conditions. In this section we optimize the conditions on both sides of the interface to get a faster convergence speed regardless
of the value of γ. By keeping the notations ζ, ζ, µ, and
q section and
q γ defined in the previous
opt
ζmin ζmax
0
p2 and λ02 =
by approximating λopt
1 and λ2 respectively by λ1 =
2
the convergence factor ρRR reads
s
(p1 − ζ)2 + ζ 2 (p2 − γζ)2 + γ 2 ζ 2
.
ρRR (p1 , p2 , ζ) =
(p1 + γζ)2 + γ 2 ζ 2 (p2 + ζ)2 + ζ 2
ζmin ζmax
2
p1 ,
We can easily demonstrate that for nonnegative fixed values of ζ and γ and for p1 , p2 > 0, we
find the three inequalities ρRR (p1 , p2 , ζ) < ρRR (−p1 , −p2 , ζ), ρRR (p1 , p2 , ζ) < ρRR (p1 , −p2 , ζ),
and ρRR (p1 , p2 , ζ) < ρRR (−p1 , p2 , ζ). This shows that we can restrict our study to strictly positive values of p1 and p2 (note that p1 = 0 or p2 = 0 corresponds to the Neumann-Robin case).
The restriction of the parameter range to strictly positive values ensures that λ01 + λ02 > 0,
and hence the corresponding problem is well-posed. In the following, we assume that γ ≥ 1.
The optimal parameters p1 and p2 for the case γ ≤ 1 can be obtained by switching (i.e. p1
becomes p2 and p2 becomes p1 ) the optimal values for the case γ ≥ 1. As it was done
previously, we choose the values p1 and p2 by solving the optimization problem
max
ρRR (p1 , p2 , ζ) .
(4.1)
min
−1
p1 ,p2 >0
ζ∈[µ
,µ]
4.1. Behavior of the convergence factor with respect to the Robin parameters. First,
we study the behavior of ρRR with respect to the parameters p1 and p2 . We introduce two new
parameters q1 and q2 defined by
q1 =
p2
p1
p
p
and q2 =
.
2
1−γ+ 1+γ
γ − 1 + 1 + γ2
We can demonstrate that for γ ≥ 1 and q1 ≤ q2 , we have that ρRR (p1 , p2 , ζ) ≤ ρRR (p2 , p1 , ζ).
This proves that the optimal parameters satisfy q1⋆ ≤ q2⋆ . This implies that in turn p1 ≤ p2
and that p1 < p2 if γ > 1. This immediately proves that one-sided (p1 = p2 ) Robin-Robin
boundary conditions are not
optimal as soon as γ > 1.
∂ρRR
RR
Note that Sign ∂p1 = Sign(q1 − ζ) and Sign ∂ρ
= Sign(q2 − ζ) implies
∂p2
(4.2)
∂ρRR
>0
∂p1
∂ρRR
>0
∂p2
when ζ < q1 ,
when ζ < q2 ,
∂ρRR
<0
∂p1
∂ρRR
<0
∂p2
when ζ > q1 ,
when ζ > q2 .
Looking at the signs of the derivatives of ρRR with respect to p1 and p2 , we find that if we
choose q1 < ζ min = µ−1 , we can decrease the convergence factor by increasing p1 beRR
cause ∂ρ
∂p1 < 0 holds for all q1 > ζ min . A similar argument shows that q2 ≤ ζ max . This
means that the optimized parameters q1⋆ and q2⋆ must satisfy
(4.3)
µ−1 ≤ q1⋆ < q2⋆ ≤ µ.
The inequalities (4.2) and (4.3) imply that at ζ = 1/µ, ρRR is an increasing function of p1
and p2 (or q1 and q2 ) while at ζ = µ, ρRR is a decreasing function of p1 and p2 (or q1 and q2 ).
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4.2. Extrema of ρRR with respect to ζ. The next step to solve (4.1) analytically is to
find the location of the extrema of ρRR (p1 , p2 , ζ, γ) with respect to ζ.
T HEOREM 4.1 (Extrema of ρRR (ζ)). The function ζ → ρRR (p1 , p2 , ζ) has one or three
positive local extrema.
q In the case of one extremum, it corresponds to a minimum and is
1 p2
located at ζ = χ := p2γ
.
Proof. We first introduce the following property that can easily be verified:
r
p1 p2
.
ρRR (p1 , p2 , ζ) = ρRR p1 , p2 , χ2 /ζ , where χ =
2γ
Differentiating with respect to ζ leads to
(4.4)
χ2 ∂ρRR
∂ρRR
(p1 , p2 , ζ) = − 2
(p1 , p2 , χ2 /ζ),
∂ζ
ζ ∂ζ
∂ρRR (p1 ,p2 ,ζ)
RR
has the same sign as a
which shows that ∂ρ
∂ζ (p1 , p2 , ±χ) = 0. The derivative
∂ζ
(unitary) sixth-order polynomial P (ζ) (the full expression of P is complicated and not given
here). P (ζ) has thus either two or six real roots, among them ζ = χ is positive and ζ = −χ
is negative. Let us suppose that P (ζ) has six real roots. We can show that only three of
these six roots (including ζ = χ) are positive. From (4.4) we see that if ζ 0 is a root of P (ζ),
then ζ 1 = χ2 /ζ 0 is another one. Assuming that the four other roots are positive, we have
ζ 5 = −χ ≤ 0 ≤ ζ 6 ≤ ζ 1 ≤ ζ 2 = χ ≤ ζ 3 (=
χ2
χ2
) ≤ ζ 4 (=
),
ζ1
ζ6
and the sum of the six roots must be greater than 2χ and is therefore positive. However, the
sum of the six roots of P (ζ) is given by −a5 where a5 is the coefficient of the term ζ 5 and is
2 −p1 )
. Using the facts γ ≥ 1 and p2 ≥ p1 (from (4.3)), −a5 cannot be
equal to a5 = (γ−1)(p
γ
positive. We conclude that we have at most three positive roots for P (ζ). It can be verified
that P (0) < 0 and P (+∞) > 0 so that if only one positive root exists (at ζ = χ), it is a local
minimum.
4.3. Equioscillation of ρRR at the end points.
T HEOREM 4.2 (Equioscillation at the end points). The optimized convergence factor ρRR (p⋆1 , p⋆2 , ζ) satisfies
q ⋆ ⋆
p1 p2
• ρRR (p⋆1 , p⋆2 , χ) ≤ max ρRR (p⋆1 , p⋆2 , µ−1 ), ρRR (p⋆1 , p⋆2 , µ) with χ =
2γ ,
• the equioscillation property ρRR (p⋆1 , p⋆2 , µ−1 ) = ρRR (p⋆1 , p⋆2 , µ), which holds only
for p⋆1 p⋆2 = 2γ.
Proof. We first show that ρRR (p⋆1 , p⋆2 , χ) ≤ max ρRR (p⋆1 , p⋆2 , µ−1 ), ρRR (p⋆1 , p⋆2 , µ) . This
RR (ζ)
is straightforward when χ is the only positive root of ∂ρ∂ζ
because χ is a local minimum. In theqcase when there are three positive roots, χ is a local maximum. From the
√
1 p2
identity χ = p2γ
= q1 q2 and (4.3), we get
(4.5)
1/µ ≤ q1 ≤ χ =
√
q1 q2 ≤ q2 ≤ µ.
We already know that at ζ = 1/µ, ρRR is a decreasing function of q1 and that at ζ = µ, ρRR
is an increasing function of q1 . The inequality (4.5) shows that at ζ = χ, ρRR is an increasing function of q1 since q1 ≤ χ. If we assume that ρRR (p⋆1 , p⋆2 , χ) ≥ ρRR (p⋆1 , p⋆2 , µ−1 ),
then we can always decrease q1 (or p1 ) such that it improves the convergence factor (by reducing the values both at ζ = χ and at ζ = µ). Playing with q2 , we can similarly prove
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that ρRR (p⋆1 , p⋆2 , χ) ≤ ρRR (p⋆1 , p⋆2 , µ). Note that this also proves that ζ 1 ≥ 1/µ and ζ 3 ≤ µ.
This is sufficient to fully describe the behavior of the convergence factor with respect to q1 , q2 ,
and ζ, as shown in Figure 4.1. In practice, the two cases will differ by the sign of the secondorder derivative of ρRR (p1 , p2 , ζ) at ζ = χ. The following argument proves that the values
taken by ρRR (p⋆1 , p⋆2 , ζ) at the two end points ζ = 1/µ, and ζ = µ are equal. Indeed, if we
assume that ρRR (p1 , p2 , 1/µ) < ρRR (p1 , p2 , µ) (or ρRR (p1 , p2 , 1/µ) > ρRR (p1 , p2 , µ)) holds, it
is always possible to decrease the maximum value of ρRR (ζ) by increasing (respectively decreasing) the values of p1 (respectively p2 ). The optimal parameters must thus satisfy the
equioscillation property ρRR (p⋆1 , p⋆2 , µ−1 ) = ρRR (p⋆1 , p⋆2 , µ). This holds when
(4.6)
(p1 + p2 )(2γ − p1 p2 )S(p1 , p2 , µ, γ) = 0,
where
S(p1 , p2 , µ, γ) = 2 (1 + γ 2 ) − γ(µ + µ−1 )2 p1 p2
+ (γ − 1)(µ + 1/µ)(p1 − p2 )(2γ + p1 p2 )
+ 2γ(p1 − p2 )2 − (2γ − p1 p2 )2 .
Obviously every pair (p1 , p2 ) that satisfies the relation p1 p2 = 2γ is a solution to (4.6). We
now show that there are no other admissible values. Other potential solutions of the problem
are the solutions of S(p1 , p2 , µ) = 0. S is a second-order polynomial in p2 and thus has two
real solutions:
(4.7)
p2 = f1 (p1 ),
p2 = f2 (p1 ).
If we assume that p2 is related to p1 with one of the relations (4.7), looking at Figure 4.1,
dp2
we can argue that for any pair (p1 , p2 ) we must have dp
< 0 to satisfy an equioscillation
1
†
property. Indeed, let ρRR (p1 , ζ) be defined as
ρ†RR (p1 , ζ) = ρRR (p1 , p2 (p1 ), ζ).
Then
(4.8)
∂ρRR (p1 , p2 (p1 ), ζ) ∂ρRR (p1 , p2 (p1 ), ζ) dp2
∂ρ†RR (p1 , ζ)
=
+
.
∂p1
∂p1
∂p2
dp1
We have already proved that the following properties must hold
(4.9)
∂ρRR (p1 , p2 (p1 ), 1/µ)
> 0,
∂p1
∂ρRR (p1 , p2 (p1 ), µ)
< 0,
∂p1
∂ρRR (p1 , p2 (p1 ), 1/µ)
> 0,
∂p2
∂ρRR (p1 , p2 (p1 ), µ)
∂p2
< 0.
dp2
If we suppose dp
> 0, then (4.8) and (4.9) show that ρ†RR (p1 , 1/µ) is an increasing function
1
of p1 while ρ†RR (p1 , µ) is a decreasing function of p1 . Hence, (4.9) and the equioscillation
dp2
property cannot be satisfied at the same time if dp
> 0. It can be shown that the two solutions
1
given by (4.7) do not satisfy this latter condition. Indeed, we can prove that we have
df1
dp1
>0
df2
and dp
> 0. Details of the computations are omitted here but we mention that the only
1
conditions necessary to find this result are γ > 0, µ > 1. We can conclude that p1 p2 = 2γ is
the only solution leading to an equioscillation
property.
q
p1 p2
It is worth mentioning that χ =
2γ = 1 and that
ρRR (p⋆1 , p⋆2 , ζ) = ρRR (p⋆1 , p⋆2 , 1/ζ),
∀ζ ∈ [1/µ, µ].
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159
F IG . 4.1. Behavior of the convergence factor with respect to ζ.
4.4. Solution of the minmax problem. The convergence factor is now a function of p1
and ζ only:
ρ†RR (p1 , ζ) = ρRR (p1 , 2γ/p1 , ζ).
L EMMA 4.3. The solution of the minmax problem (4.1) is given by the solution of the
constraint minimization problem
min
⋆,equi
p⋆
1 ≥p1
ρ†RR (p⋆1 , 1/µ),
where p⋆,equi
is the solution of the three-point equioscillation problem
1
ρ†RR (p1 , 1) = ρ†RR (p1 , 1/µ) = ρ†RR (p1 , µ).
Proof. Thanks to Figure 4.1, we can remark that the resolution of the minmax problem
corresponds to the minimization of ρ†RR (p1 , 1/µ) (or ρ†RR (p1 , µ)) with respect to p1 . In the
case where χ = 1 is a local maximum, the additional constraint given by Theorem 4.2 must
be imposed
(4.10)
ρ†RR (p1 , 1) ≤ ρ†RR (p1 , 1/µ).
Knowing that p1 p2 = 2γ or equivalently q1 q2 = 1, the range of admissible values given
by the inequality (4.3) can now be written as 1/µ ≤ q1 ≤ 1 and translates in terms of the
variable p1 :
(4.11)
p1 ∈ [p1,min , p1,max ], where
p
p1,min = (1 − γ + 1 + γ 2 )/µ,
p1,max = (1 − γ +
p
1 + γ 2 ).
Moreover, it can be shown that ρ†RR (p1 , 1) is a decreasing function of p1 and therefore the
constraint (4.10) can also be written as p⋆1 ≥ p⋆,equi
where p⋆,equi
is the solution of a three1
1
⋆,equi
⋆,equi
†
†
, µ)).
, 1/µ)(= ρ†RR (p⋆,equi
, 1) = ρRR (p1
point equioscillation problem ρRR (p1
1
We now look at the minimization of ρ†RR (p1 , 1/µ) for p1 ∈ [p1,min , p1,max ].
L EMMA 4.4. For γ > 1, the derivative of ρ†RR (p1 , 1/µ) has exactly one root in the
range [p1,min , p1,max ]. This root corresponds to a local minimum of ρ†RR (p1 , 1/µ). In the
√
∂ρ†
special case γ = 1, p1 = p1,max (= 2) is always a root of ∂pRR1 (p1 , 1/µ).
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F. LEMARIÉ, L. DEBREU, AND E. BLAYO
F IG . 4.2. Behavior of ρ†RR (p1 , 1/µ) with respect to p1 . The general case (γ > 1) is on the left and the special
case γ = 1 on the right.
Proof. The derivative of ρ†RR (p1 , 1/µ) can be written as
∂ρ†RR
(p1 , 1/µ) = g(p1 , µ)N (p1 , µ),
∂p1
where g is a strictly positive function and N (p1 , µ) a sixth-order polynomial in p1 . The
change of variable v = 2γ/p1 − p1 transforms N (p1 , µ) into
N (p1 , µ) = p31 Q(v),
where Q(v) is the third-order polynomial given by
(4.12)
Q(v) = 8(γ − 1)(1 + γ 2 ) + 2β(γβ 2 − 3(1 + γ 2 ))v + 2(γ − 1)β 2 v 2 − βv 3 ,
with β = 1/µ + µ.
It can be shown that, for γ > 1, this polynomial has only one root in [vmin , vmax ], where,
according to (4.11), vmin and vmax are given by
p
p
vmin = 2(γ − 1), vmax = (γ − 1)β + 1 + γ 2 β 2 − 4.
This root corresponds to a minimum of the functional ρ†RR (p1 , 1/µ) because we can show
∂ρ†
∂ρ†
that ∂pRR
(p1,min , 1/µ) ≤ 0 and ∂pRR
(p1,max , 1/µ) ≥ 0. For γ = 1, the value v = vmin = 0,
1
1
√
i.e., p1 = p1,max = 2, is always a root of Q(v). Figure 4.2 illustrates the variations
of ρ†RR (p1 , 1/µ) with respect to p1 . pmin
is the location of the minimum of ρ†RR (p1 , 1/µ)
1
over the interval [p1,min , p1,max ]. The solution of the constrained minimization problem is
now easily handled: if pmin
≤ p⋆,equi
, then the solution of the minmax problem is given
1
1
⋆,equi
by p1
, otherwise the solution of the minmax problem is given by pmin
1 .
∂ρ†RR
⋆,equi
, µ)
∂p1 (p1
p⋆,equi
.
1
The inequality pmin
≤ p⋆,equi
is satisfied if and only if
1
1
≥ 0 or equiva-
lently if Q(v ⋆,equi ) ≥ 0, where v ⋆,equi = 2γ/p⋆,equi
−
1
Finally, the following result is useful: for v ≥ vmax (or equivalently p1 ≤ p1,min ), we
∂ρ† (p ,1/µ)
have Q(v) ≤ 0 or RR ∂p11
≤ 0 . Indeed, using the relation (4.8) at ζ = 1/µ, we obtain
∂ρRR (p1 , p2 (p1 ), 1/µ) ∂ρRR (p1 , p2 (p1 ), 1/µ) dp2
∂ρ†RR (p1 , 1/µ)
=
+
.
∂p1
∂p1
∂p2
dp1
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161
1 ),1/µ)
If p1 ≤ p1,min holds, then we get ∂ρRR (p1 ,p∂p2 (p
< 0, but the relation p2 = 2γ/p1 implies
1 p
1 ),1/µ)
≥ 0. With the help
that p2 ≥ p2,max = γ − 1 + 1 + γ 2 µ so that ∂ρRR (p1 ,p∂p2 (p
2
∂ρ† (p ,1/µ)
dp2
dp1
= −2γ/p21 ≤ 0, this proves that RR ∂p11
≤ 0.
We are now finished with the problem of finding the solution of the three-point equioscillation problem.
T HEOREM 4.5 (Equioscillation between three points). The only parameters p⋆,equi
1
and p⋆,equi
, such that p⋆,equi
≤ p1,max , that satisfy an equioscillation of the convergence
2
1
factor ρRR between the three points (1/µ, 1, µ) are
of
i
1 h ⋆,equi p
+ 8γ + (v ⋆,equi )2 ,
−v
2
−1
= 2γ p⋆,equi
,
1
p⋆,equi
=
1
p⋆,equi
2
where
i
p
1h
(2 + β)(γ − 1) + 4(1 + γ)2 (β − 1) + β 2 (γ − 1)2 .
2
Proof. We have to find the solution of the problem ρ†RR (p1 , 1/µ) = ρ†RR (p1 , 1). It can be
shown that this is equivalent to the search for the roots of a fourth-order polynomial R(p1 )
that can be written as
v ⋆,equi =
(4.13)
R(p1 ) = p21 T (v),
T (v) = 2(1 + γ 2 ) − 4γβ + (1 − γ)(2 + β)v + v 2 ,
where v is again defined by v = 2γ/p1 − p1 . The unique root of T (v) that satisfies v ≥ vmin
(i.e., p1 ≤ p1,max ) is given by
v ⋆,equi =
i
p
1h
(2 + β)(γ − 1) + 4(1 + γ)2 (β − 1) + β 2 (γ − 1)2 ,
2
and the expression for p⋆,equi
is deduced from the relation between p1 and v.
1
Gathering the results, the solution of the minmax problem is given in the following theorem.
0,⋆
T HEOREM 4.6. The analytical solution λ0,⋆
1 and λ2 of the minmax problem
0
0
max
ρRR (λ1 , λ2 , D1 , D2 , ω)
min
0
0
λ1 ,λ2 ∈R
ω∈[ωmin ,ωmax ]
is given by
√
1/4 h
i
p
D1 (ωmin ωmax )
√
−v ⋆ + 8γ + (v ⋆ )2 ,
2 2
p
√
= D1 D2 ωmin ωmax /λ0,⋆
1 ,
λ0,⋆
1 =
λ0,⋆
2
where
⋆
v =
(
v ⋆,equi
v ⋆,mini
if Q(v ⋆,equi ) ≥ 0,
else,
with v ⋆,equi given by (4.13) and v ⋆,mini is the unique solution of Q(v) = 0 in [vmin , vmax ].
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F. LEMARIÉ, L. DEBREU, AND E. BLAYO
F IG . 4.3. Transition from a two-point to a three-point equioscillation for β 0 < β < 1 +
equioscillation occurs when γ ≥ f (β).
√
5. The three-point
Proof. All the ingredients for the proof are stated before. Note that v ⋆,equi may be larger
than vmax . However, since we have proved that Q(v ≥ vmax ) ≤ 0, this case does not have
to be considered explicitly. A p
substitution of γ and µ by their respective expressions, and a
0,⋆
multiplication of p⋆1 and p⋆2 by ζmin ζmax /2 lead to the result for λ0,⋆
1 and λ2 with respect
to D1 , D2 , ωmin , and ωmax .
Note that the following additional result can be shown as well:
√
√
(4.14)
Q(v ⋆,equi ) ≥ 0 ⇔ β ≥ 1 + 5 or β 0 < β < 1 + 5 and γ ≥ f (β) ,
where β 0 is the root of the fourth-order polynomial 16−16X −4X 2 +X 4 whose approximate
value is given by β 0 ≈ 2.77294 and f is given by
f (β) =
1
(β − 2)3 β(β + 2)
16 − 16β − 4β 2 + β 4
p
+ (4 + 2β − β 2 ) −16 + 48β − 44β 2 + 12β 3 + 3β 4 − 4β 5 + β 6 .
√
The function f (β) is plotted in Figure 4.3 for β 0 < β < 1 + 5. We remark that f (β) ≥ 1
for all β so that the condition γ ≥ f (β) is always
false for γ = 1 (the continuous case).
q
p1 p2
It is also interesting to know if χ =
2γ = 1 is either a local minimum or a local
∂ 2 ρ†RR
∂χ2 (p1 , χ). It
∂ 2 ρ†RR
∂χ2 (p1 , χ) > 0
maximum of the optimized convergence factor by looking at the sign of
can
be proved that in terms of the variable v = 2γ/p1 − p1 , the inequality
be written as
p
v ≥ v0 , where v0 = 2(γ − 1) + 2(1 + γ 2 ).
can
We deduce that ζ = χ = 1 is a local minimum only if v ⋆,mini ≤ v0 . This can be verified by
evaluating the polynomial Q(v) at v = v0 and looking at the sign of the result: if Q(v0 ) ≤ 0,
then v ⋆,mini ≤ v0 and we have a local minimum at ζ = χ = 1.
It can be found that
√
Q(v0 ) < 0 ⇔ 2 < β < β0
or β0 ≤ β ≤ 2 2 and γ < g(β) ,
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F IG . 4.4. The three different domains of three-point equioscillation (black), two-point equioscillation with χ
being a local maximum (dark grey) and two-point equioscillation with χ being a local minimum (light grey).
F IG . 4.5. Optimized convergence factor with respect to µ and γ (1 ≤ µ ≤ 10, 1 ≤ γ ≤ 10).
where β0 =
√
8+5 √2
2(3+2 2)
+
√
√
90+64 2
√
2(3+2 2)
≈ 2.44547. The analytical expression of g(β) is com-
plicated and not given here. Note that g(β) ≥ 1 for √
all β so that for the special case γ = 1,
the inequality Q(v0 ) < 0 is equivalent to 2 < β ≤ 2 2.
Figure 4.4 summarizes the three different domains: three-point equioscillation, two-point
equioscillation with χ as a local maximum and two-point equioscillation with χ as a local
minimum. The resulting optimized convergence factor is shown in Figure 4.5 with respect
to µ and γ.
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F. LEMARIÉ, L. DEBREU, AND E. BLAYO
F IG . 4.6. Optimized convergence factors for Neumann-Robin and Robin-Robin boundary conditions for µ = 2
(left) and µ = 6 (right).
We make the following remarks about the convergence properties of the Schwarz algorithms: the convergence speed increases when the discontinuities of the coefficients (γ)
is increased and the convergence speed decreases when µ, an increasing function of the
ratio ωωmax
, is increased. In Figure 4.6 we compare, for µ = 2 and µ = 6, the results
min
found in the optimized two-sided case with the optimized Robin-Neumann transmission conditions (found in Section 3). The Robin-Robin approach is significantly more efficient than
the Robin-Neumann approach when γ is close to one. When γ is increased, both tend towards
a Dirichlet-Neumann operator.
π
, and
T HEOREM 4.7 (Asymptotic performance). For D2 > D1 (i.e., γ > 1), ωmax = ∆t
for ∆t tending to zero, the optimal Robin parameters given by Theorem 4.6 are
p
γ √
γ2 + 1
3/4
0,(as)
1/4
−
2γ
,
ω
∆t
λ0,⋆
≈
λ
=
2D
ω
1
min
1
1
γ−1
(γ − 1)3 π 1/4 min
p
γ − 1√
γ2 + 1
0,(as)
1/4
−1/4
−1/2
λ0,⋆
≈
λ
=
,
(πω
)
∆t
2D
π∆t
+
1
min
2
2
2
γ−1
and the asymptotic convergence of the optimized two-sided Robin-Robin algorithm is
(γ + 1) ωmin 1/4 1/4
1
0,⋆
0,⋆
max π ρRR (λ1 , λ2 , ω) =
1−2
+ O(∆t1/2 ).
∆t
ωmin ≤ω≤ ∆t
γ
(γ − 1)
π
Note that these asymptotic results are obtained by assuming that v ⋆ = v ⋆,equi , which is
always the case when ∆t → 0 (i.e., µ → ∞), as shown by p
(4.14). The optimized Robin
Robin conditions lead to an asymptotic convergence factor D1 /D2 1 − O(∆t1/4 ) for
small ∆t and D1 < D2 . The associated algorithm is thus less sensitive to ∆t than the
Neumann-Robin algorithm. However, the asymptotic Robin parameters given in Theorem 4.7
must be used with caution as they degenerate when γ → 1 as well as when ∆t ≫ 0 (in this
0,(as)
case λ1
can become negative). It is worth mentioning that the asymptotic bound on the
optimized convergence factor given in Theorem 4.7 shows that the optimized Robin-Robin
conditions will always be more efficient than Dirichlet-Neumann conditions. Indeed, it can
easily be checked that the multiplicative factor 1/γ in front of the bound corresponds to the
convergence factor of the Dirichlet-Neumann algorithm.
Furthermore, we can not directly compare this result with the one obtained in [10] for
the advection-diffusion-reaction equation. The latter study is done by assuming ωmin = 0
and as a result of this assumption their optimized parameters, when canceling the advection
and reaction coefficients, are simply λ0,⋆
= λ0,⋆
= 0. Indeed, one can easily find that for
1
2
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165
opt
a diffusion problem, the low frequency approximation λlow
j of the absorbing conditions λj
low
given in (2.7) for ωmin → 0 is indeed λj = 0.
4.5. The continuous case. Because the two-sided Robin-Robin case with continuous
diffusion coefficients has never been studied in the literature, we now provide the results in
this particular case.
T HEOREM 4.8 (Continuous case). Under the assumption D1 = D2 = D, the optimal
0,⋆
parameters λ0,⋆
1 and λ2 are given by
√
1/4 h
i
p
D (ωmin ωmax )
0,⋆
√
λ1 =
−v ⋆ + 8 + (v ⋆ )2
2 2
√
1/4 h
i
p
D (ωmin ωmax )
√
λ0,⋆
v ⋆ + 8 + (v ⋆ )2
2 =
2 2
where
with β =
√
ωmax +
 p

2 β−1



p
v⋆ =
2β 2 − 12



0
√
if β ≥ 1 + 5
√
√
if 6 ≤ β < 1 + 5
√
if 2 < β < 6
√
ωmin
.
1/4
(ωmin ωmax )
Proof. We use Theorem 4.6, which gives the optimality conditions√in the general case.
As already mentioned, the condition Q(v ⋆,equi ) ≥ 0 reduces to β ≥ 1 + 5 for γ = 1.
p In that
⋆
⋆,equi
case, the √
solution of the minmax problem is given by v = v
= 2 β − 1.
⋆,min
If β < 1 + 5, we have to
compute
v
,
the
value
that
cancels
Q(v)
in
[vmin , vmax ],
p
where vmin = 0, vmax = 2 β 2 − 4. For γ = 1, the expression (4.12) of the polynomial Q(v)
is
Q(v) = −βv v 2 − (2β 2 − 12) .
We find that
v ⋆,min =
p
0,
2β 2 − 12,
√
if β ≥ 6, √
if 2 < β ≤ 6.
p
√
1/4
0,⋆
Note that when β ≤ 6, we get v ⋆ = 0. This implies λ0,⋆
D1 (ωmin ωmax ) ,
1 = λ2 =
which corresponds to the zeroth-order one-sided optimal parameters found in [8].
5. Numerical experiments with two subdomains. The model problem (2.2) is discretized using a backward Euler scheme in time and a second-order scheme on a staggered
grid in space. For the interior points, the scheme is
i
h
un+1
− unk
1
n+1
k
=
,
Fn+1
1
1 − F
k− 2
∆t
xk+ 12 − xk− 21 k+ 2
with Fn
1
k+ 2
= D
n
un
k+1 −uk
.
1
k+ 2 xk+1 −xk
Note that for practical applications, the use of the Crank-
Nicolson scheme in time is avoided because this leads to unphysical behavior. Indeed, unlike the backward Euler scheme, the Crank-Nicolson scheme fails to satisfy the so-called
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F. LEMARIÉ, L. DEBREU, AND E. BLAYO
monotonic damping property [22]. We decompose the computational domain Ω into two
non-overlapping subdomains Ω1 = [−L1 , 0] and Ω2 = [0, L2 ], with L1 = L2 = 500 m.
A homogeneous Neumann boundary condition is imposed at x = −L1 and x = L2 . As it
is usually done in numerical models, the resolution ∆xk is progressively refined to enhance
the resolution in the boundary layers in the vicinity of the air-sea interface. We use N = 75
points in each subdomain and the resolution varies from ∆xk = 25 m at x = L1 (respectively x = L2 ) to ∆xk = 1 m at x = 0. The Robin condition gN + 12 on the interface Γ
(located at x = xN + 21 on Ω1 and at x = x 12 on Ω2 ) is discretized by assuming that the flux F
is constant on the first cell near Γ. This leads to
gN + 21 = DN − 12
uN − uN −1
+ λuN ,
xN − 21 − xN − 32
where λ is the Robin parameter. We simulate directly the error equations, i.e., f1 = f2 = 0
in (2.2) and u0 (x) = 0. We start the iteration with a random initial guess u02 (0, t), t ∈ [0, T ],
so that it contains a wide range of the temporal frequencies that can be resolved by the computational grid. We perform simulations for four different types of transmission conditions
at x = 0: Dirichlet-Neumann (DN), optimized Neumann-Robin (NR⋆ ), optimized RobinRobin (RR⋆ ), and asymptotically optimized Robin-Robin (RR(as) ). In Figure 5.1 we show the
1
∞
evolution
√ of the L -norm of the error obtained for those four cases for γ = 10 4 ≈ 1.7783,
γ = 10 ≈ 3.1623, and γ = 10, with µ = 6 and µ = 12. We choose the time
steps ∆t1 = ∆t2 = ∆t = 100 s, D2 = 0.5 m2 s−1 , and D1 is then deduced depending
on the value of γ. As expected, we get the best results with the two-sided Robin conditions. Consistent with Figure 4.5, the convergence is faster when γ is large and when µ is
small. Moreover, when the discontinuity γ between the diffusion coefficients is increased,
the algorithm becomes less and less sensitive to the choice of transmission conditions and
to the parameter µ. The asymptotic optimized Robin-Robin conditions provide a good approximation of the optimized Robin-Robin conditions, even for ∆t = 100 s ≫ 0. Those
conditions are especially efficient when γ is sufficiently larger than 1. We remark that the
optimized Neumann-Robin conditions provide only a slight improvement compared to the
classical Dirichlet-Neumann conditions.
Because we consider a problem with discontinuous coefficients, the time step used to
advance the diffusion equation may be different in each subdomain. It is thus instructive to
look at the impact of non-conformities in time on the performance of the optimized algorithm.
1
For the two cases γ = 10 4 and γ = 10, which we considered so far, we adapt the time step
in each subdomain so that the ratio r = ∆t1 /∆t2 between the time steps varies from 100
to 1/100. To handle the exchange of boundary data between the non-conforming grids we
use a linear method for the interpolation step and an averaging for the restriction step, both
steps are conservative. Note that we got very similar results using the L2 projection algorithm
described in [13, Appendix A]. We consider ωmax = π/ min(∆t1 , ∆t2 ) for the optimization
of the Robin conditions.
As shown in Figure 5.2, the performance of the algorithm is degraded as long as r 6= 1.
Indeed, the interpolation/restriction step modifies the frequency spectrum of the error and thus
affects the convergence speed of the algorithm. For the cases r 6= 1, we have investigated a
wide range of values for the parameters pj in the Robin transmission conditions. Even if the
algorithm is slower than the one with r = 1, it turned out that the optimal Robin parameters
found in Theorem 4.6 are still optimal in the case r 6= 1. Note that the results after one
iteration can be quite dependent on the value of r. This is to be expected since the frequency
spectrum in the initial error is very different whether the (random) initial guess is initialized
on the grid with the smaller time step or the larger time step.
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167
√
1
F IG . 5.1. Convergence for γ = 10 4 (top, left), γ = 10 (top, right), and γ = 10 (bottom, left) for µ = 6
⋆
⋆
and µ = 12 in the DN, RR , and NR cases. Comparison between RR⋆ and RR(as) (bottom, right).
1
F IG . 5.2. Convergence for γ = 10 4 (left) and γ = 10 (right) for different values of the ratio r = ∆t1 /∆t2
with µ = 6 in the RR⋆ case.
Conclusion. In this paper, we obtain new results for an optimized Schwarz method defined for non-overlapping diffusion problems with discontinuous coefficients. This method
uses zeroth-order two-sided Robin transmission conditions, i.e., we consider two different
Robin conditions on each side of the interface. We base our approach on a model problem
with two subdomains and we prove the convergence of the corresponding algorithm. Then
we analytically study the behavior of the convergence factor with respect to the parameters
of the problem. We show that the optimized convergence factor satisfies an equioscillation
property between two or three points depending on the parameter values. In comparison with
other methods using the Neumann-Robin or Dirichlet-Neumann conditions, these two-sided
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F. LEMARIÉ, L. DEBREU, AND E. BLAYO
Robin-Robin conditions are significantly more efficient, especially when the ratio between the
discontinuous coefficients is close to one. Asymptotic results for ∆t small are given. Numerical results show the performance of the different type of transmission conditions introduced
in this paper. Those results are consistent with the analytical study.
Acknowledgements. This research was partially supported by the French National Research Agency (ANR) project ”COMMA” and by the INRIA project-team MOISE. The authors would also like to thank two anonymous reviewers, whose comments and suggestions
during the review process helped to clarify earlier version of the manuscript.
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